Measure Theory

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Properties of exponantiation of ordinals

In the last post, we worked pretty hard to define ordinals exponentiation, however, This hard work revealed some great properties: If is a successor then: If is a limit ordinal then; Those properties are great – the first two are pretty intuitive, and the third is also a pretty comfortable definition of . We canContinue reading “Properties of exponantiation of ordinals”

The Cyclotomic polynomial

After we’ve met the terms of roots of unity, primitive roots, and deduced a great identity, we are now finally ready to define the Cyclotomic polynomial of order . It is the polynomial where it’s roots are the primitive roots of unity of order : Let’s find out what’s so special about it: First, noteContinue reading “The Cyclotomic polynomial”

Roots of unity

After we’ve finally proved The Fundametal theorem of Galois Theory, we can officially say that we understand the correspondence, and we know exactly when it behaves as we want it to behave. So, it’s time to see some results! The first topic I want to discuss is related to the roots of unity. We’ve metContinue reading “Roots of unity”

The Fundamental Theorem of Galois Theory

After meeting the term of a Galois extension and Artin’s lemma, we are ready to prove the fundamental theorem of galois theory! Recall that when is a galois extension, we can say a lot about the extension: is a normal and separable. is a splitting field of a separable polynomial. is a fixed field, thatContinue reading “The Fundamental Theorem of Galois Theory”


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